Inverse Laplace Transform Theorem
Linearity Theorem
If a and b are constants, and F(s) and G(s) are the Laplace transforms of functions f(t) and g(t) respectively, then the inverse Laplace transform of aF(s) + bG(s) is given by:
L-1 {aF(s)+bG(s)}=a.L-1 {F(s)}+b.L-1 {G(s)}
To find the inverse Laplace transform of a linear combination of transformed functions, you can use this theorem. Simply take a weighted sum of their respective inverse Laplace transforms.
Shifting Theorem
If F(s) is the Laplace transform of a function f(t), then the inverse Laplace transform of e-at F(s)is given by:
L-1 {e-atF(s)} = f(t-a)
This Property illustrates how multiplying the Laplace transform by e-at in the Laplace domain corresponds to shifting the original function f(t) by a units to the right in the time domain.
Convolution Theorem
The Convolution Theorem for Laplace Transforms states that if F(s) and G(s) are the Laplace transforms of functions f(t) and g(t) respectively, then the Laplace transform of their convolution, denoted as f(t) × g(t), is equal to the product of their individual Laplace transforms. Mathematically, it can be expressed as:
L{f(t) × g(t)}=F(s)⋅G(s)
To find the Laplace transform of two functions convolved in the time domain, you can multiply their individual Laplace transforms in the s-domain.
Inverse Laplace Transform
In this Article, We will be going through the Inverse Laplace transform, We will start our Article with an introduction to the basics of the Laplace Transform, Then we will go through the Inverse Laplace Transform, will see its Basic Properties, Inverse Laplace Table for some Functions, We will also see the Difference between Laplace Transform and Inverse Laplace Transform, At last, we will conclude our Article with Some examples of inverse Laplace Transform, Applications of inverse Laplace and Some FAQs.
Table of Content
- Inverse Laplace Transform
- Inverse Laplace Transform Theorem
- Inverse Laplace Transform Table
- Laplace Transform Vs Inverse Laplace Transform
- Properties
- Advantages and Disadvantages
- Applications
- Examples